Let p: ‘If x is a real number such that x3 + 4x = 0, then x is 0’
q: x is a real number such that x3 + 4x = 0
r: x is 0
(i) We assume that q is true to show that statement p is true and then show that r is true
Therefore, let statement q be true
Hence, x3 + 4x = 0
x (x2 + 4) = 0
x = 0 or x2 + 4 = 0
Since x is real, it is 0.
So, statement r is true.
Hence, the given statement is true.
(ii) By contradiction, to show statement p to be true, we assume that p is not true.
Let x be a real number such that x3 + 4x = 0 and let x ≠ 0
Hence, x3 + 4x = 0
x (x2 + 4) = 0
x = 0 or x2 + 4 = 0
x = 0 or x2 = -4
However x is real. Hence, x = 0, which is a contradiction since we have assumed that x ≠ 0
Therefore, the given statement p is true.
(iii) By contrapositive method, to prove statement p to be true, we assume that r is false and prove that q must be false
∼r: x ≠ 0
Clearly, it can be seen that
(x2 + 4) will always be positive
x ≠ 0 implies that the product of any positive real number with x is not zero.
Now, consider the product of x with (x2 + 4)
∴ x (x2 + 4) ≠ 0
x3 + 4x ≠ 0
This shows that statement q is not true.
Hence, proved that
∼r ⇒ ∼q
Hence, the given statement p is true.
Answered by Abhisek | 1 year agoDetermine whether the argument used to check the validity of the following statement is correct: p: “If x2 is irrational, then x is rational.” The statement is true because the number x2 = π2 is irrational, therefore x = π is irrational.
Which of the following statements are true and which are false? In each case give a valid reason for saying so
(i) p: Each radius of a circle is a chord of the circle.
(ii) q: The centre of a circle bisect each chord of the circle.
(iii) r: Circle is a particular case of an ellipse.
(iv) s: If x and y are integers such that x > y, then – x < – y.
(v) t: \( \sqrt{11}\) is a rational number.
By giving a counter example, show that the following statement is not true. p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”
Show that the following statement is true “The integer n is even if and only if n2 is even”
Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x2 is odd, then x is also odd.”